Problem 3 - Single Variable Calculus

Question

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{1}^{2} x^{3} d x, \quad n=6 $$

Step-by-Step Solution

Answer

The exact value of the integral \(\int_{1}^{2} x^{3} dx\) is \(\frac{15}{4} \approx 3.75\). Using the Trapezoidal Rule with \(n=6\), we get an approximation of 3.7396. Using Simpson's Rule with \(n=6\), we get an approximation of 3.7500. Comparing the results, we can observe that Simpson's Rule provides a more accurate approximation to the exact value compared to the Trapezoidal Rule.

Step 1: Compute the exact value of the integral

First, let's find the exact value of the integral \(\int_{1}^{2} x^{3} dx\). This will give us a reference for comparison with our approximations. To find the exact value, we will use the antiderivative of \(x^3\), which is \(F(x)=\frac{x^4}{4}\). Applying the Fundamental Theorem of Calculus, we have: Exact Value = \(F(2)-F(1) = \frac{2^4}{4} - \frac{1^4}{4} = \frac{16}{4} - \frac{1}{4} = \frac{15}{4}\)

Step 2: Apply the Trapezoidal Rule

Now, let's apply the Trapezoidal Rule to approximate the integral: Trapezoidal Rule = \(\frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]\) First, find the width of the interval \(\Delta x = \frac{b-a}{n}\): \(\Delta x = \frac{2-1}{6} = \frac{1}{6}\) Next, we construct a table of function values and the corresponding summands of the Trapezoidal Rule: 1. \(x_0 = 1, f(x_0) = 1^3 = 1\) 2. \(x_1 = \frac{7}{6}, f(x_1) = \left(\frac{7}{6}\right)^3 = \frac{343}{216}\) 3. \(x_2 = \frac{4}{3}, f(x_2) = \left(\frac{4}{3}\right)^3 = \frac{64}{27}\) 4. \(x_3 = \frac{3}{2}, f(x_3) = \left(\frac{3}{2}\right)^3 = \frac{27}{8}\) 5. \(x_4 = \frac{5}{3}, f(x_4) = \left(\frac{5}{3}\right)^3 = \frac{125}{27}\) 6. \(x_5 = \frac{11}{6}, f(x_5) = \left(\frac{11}{6}\right)^3 = \frac{1331}{216}\) 7. \(x_6 = 2, f(x_6) = 2^3 = 8\) Applying the Trapezoidal Rule: Approximation using Trapezoidal Rule = \(\frac{1}{12}\left[1 + 2(\frac{343}{216} + \frac{64}{27} + \frac{27}{8} + \frac{125}{27} + \frac{1331}{216}) + 8\right] \approx 3.7396\)

Step 3: Apply Simpson's Rule

Now, let's apply Simpson's Rule to approximate the integral: Simpson's Rule = \(\frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n)]\) We already have \(\Delta x = \frac{1}{6}\), and the table of function values. Applying Simpson's Rule: Approximation using Simpson's Rule = \(\frac{1}{18}\left[1 + 4\left(\frac{343}{216} + \frac{27}{8} + \frac{1331}{216}\right) + 2\left(\frac{64}{27} + \frac{125}{27}\right) + 8\right] \approx 3.7500\)

Step 4: Compare the results

Now that we have all approximations and the exact value, let's compare them: 1. Exact Value of the integral = \(\frac{15}{4} \approx 3.75\) 2. Trapezoidal Rule approximation = 3.7396 3. Simpson's Rule approximation = 3.7500 From these values, we can observe that Simpson's Rule provides a more accurate approximation (3.7500) to the exact value (3.75) compared to the Trapezoidal Rule (3.7396).

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a simple method for approximating the definite integral of a function. It works by estimating the area under the curve as a series of trapezoids, hence the name. This technique is particularly useful when dealing with functions that are difficult to integrate analytically. The formula for the Trapezoidal Rule is given by:\[\text{Trapezoidal Rule} = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]\]Key points about the Trapezoidal Rule:

• \( \Delta x \) is the width of each subinterval, calculated as \( \frac{b-a}{n} \), where \( a \) and \( b \) are the limits of integration.
• Each trapezoid's area is calculated and summed to estimate the total integral.
• The method tends to slightly underestimate or overestimate the integral, depending on the function's concavity.

In practice, the more intervals you use, the closer the Trapezoidal Rule will get to the exact integral value. However, for some functions, other methods such as Simpson's Rule might be more accurate.

Simpson's Rule

Simpson's Rule is an advanced technique for numerical integration, and it provides a more accurate approximation than the Trapezoidal Rule by fitting parabolas to segments of the curve. This method assumes that the function can be approximated by quadratic polynomials. The Simpson's Rule formula is:\[\text{Simpson's Rule} = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n)]\]Important aspects of Simpson's Rule include:

• It requires the number of intervals, \( n \), to be even to fit the quadratic approximation accurately.
• This rule is particularly effective for functions that are either quadratic or can be closely approximated by a quadratic polynomial.
• In our example, Simpson's Rule provided a closer approximation (3.7500) to the exact integral (3.75) than the Trapezoidal Rule.

Simpson's Rule is preferred when a high degree of accuracy is needed, especially when the function behaves well (e.g., smooth and continuous) over the interval.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of differentiation and integration, two main operations in calculus. It has two parts, but for our purposes, the part that helps determine the exact integral value is critical.This theorem asserts that if \( f \) is continuous over the interval \([a, b]\), and \( F \) is an antiderivative of \( f \), then:\[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]Here's how it works:

• Identify an antiderivative \( F \) of the function \( f(x) \).
• Evaluate \( F(x) \) at the upper and lower limits of the interval.
• Subtract \( F(a) \) from \( F(b) \) to find the total area under the curve.

In our specific case, this theorem efficiently provided the exact integral value, \( \frac{15}{4} \), serving as a benchmark for numerical approximations.

Antiderivative

An antiderivative of a function \( f(x) \) is another function \( F(x) \) such that the derivative of \( F(x) \) returns \( f(x) \). In simpler terms, finding an antiderivative is essentially reversing the process of differentiation.To find an antiderivative, you can often follow these steps:

• Integrate the function analytically. For example, the antiderivative of \( x^3 \) is \( \frac{x^4}{4} \).
• Apply a constant of integration if looking for a general solution. However, for definite integrals, the constant is not needed.
• Use this antiderivative as a tool for calculating exact integral values, often in conjunction with the Fundamental Theorem of Calculus.

The main importance of antiderivatives lies in their application in finding exact areas under a curve, unlike numerical methods that approximate this area.

Exact Integral Value

In calculus, the exact integral value is the precise area under the curve of a function over a specified interval. The process of determining this value involves finding an antiderivative of the function and applying the Fundamental Theorem of Calculus.For the integral \( \int_{1}^{2} x^3 \, dx \), the exact value is calculated using:\[F(2) - F(1),\text{ where } F(x) = \frac{x^4}{4}\]Calculating gives:

• \( F(2) = \frac{16}{4} = 4 \)
• \( F(1) = \frac{1}{4} \)
• Exact Integral Value = \( 4 - \frac{1}{4} = \frac{15}{4} \approx 3.75 \)

This exact value is important as it serves as the benchmark for judging the accuracy of numerical approximations such as the Trapezoidal and Simpson's Rule.